GROUPS OF HOMOMORPHISM GRADED BY G-SETS

Andrei Marcus and Ciprian Modoi

”Babes-Bolyai” University Faculty of Mathematics and Computer Science Str. Mihail Kogălniceanu nr. 1 RO-3400 Cluj-Napoca, ROMANIA

1. Introduction

Let G be a group and R = ⨁_{g∈G} R_{g} a G-graded ring. If M = ⨁_{g∈G} M_{g} and N = ⨁_{g∈G} N_{g} are G-graded rings then E.C. Dade showed in [D1] that the group Hom_{R}(M, N) has a subgroup denoted HOM_{R}(M, N) which can be endowed with a natural G-grading:

HOM_{R}(M, N)_{g} = {f ∈ Hom_{R}(M, N) | f(M_{h}) ⊆ N_{hg} for all h ∈ G}
= Hom_{R−gr}(M, N(g)),

where R-Gr denotes the category of G-graded R-modules and grade preserving Rmorphisms and N(g) is the G-graded module, called the g -suspension of N, with N(g) = N and N(g)_{h} = N_{hg} for all h ∈ G.

In particular, END_{R}(M) = HOM_{R}(M, M) is a G-graded ring and HOM_{R}(M, N) becomes a G-graded E -module, where E = END_{R}(M)^{opp} (with grading given by E_{g} = END_{R}(M)_{g^{−1}} ).

The natural problem which arises is to give condition under which HOM_{R}(M, N) = Hom_{R}(M, N). Some condition were established by Dade in [D1] and a general answer was given by Gómez-Pardo, Militaru and Năstăsescu in [GMN], using among other things the fact, proved in [GMN, Theorem 1.2], that Hom_{R}(M, N) is the completion of HOM_{R}(M, N) in the finite topology.

Dade also introduced in [D2] a useful generalization of G-graded modules, namely modules graded by G-sets. We shall be interested only in modules graded by transitive G-sets, so let H be a subgroup of G and denote by G/H the set {gH | g ∈ G} of left cosets of H in G. An R-module N is G/H -graded if N = ⨁_{x∈G/H} N_{x} (as additive subgroups) and R_{g}N_{x} ⊆ N_{gx} for all g ∈ G and x ∈ G/H. Denote by (G/H, R)-gr the category of G/H -graded R-modules and grade preserving R-morphisms. Observe that (G/G, R)-Gr=R-mod, while (G/1, R)-Gr = R-Gr. Moreover, (G/H, R)-Gr is a Grothendieck category and it was investigated in detail in [NRV] and [NSV].

ANDREI MARCUS AND CIPRIAN MODOI

The construction of “HOM” was generalized in [M, Section 2]. If M is a G-graded R-module and N is a G/H -graded R-module then Hom_{G/H,R}(M, N) has a subgroup HOM_{R}(M, N) endowed with a natural G/H -grading such that Hom_{G/H,R}(M, N) is a G/H -graded R-module.

Recall that if K ≤ H ≤ G then there is a functor

U = U_{G/K}^{G/H}: (G/K, R)-Gr → (G/H, R)-Gr

sending a G/K -graded R-module N to the G/H -graded R-module U(N) = N with

U(N)_{x} = ⨁_{\begin{matrix}y∈G/K \\ y⊆x \end{matrix}} N_{y}.

The details on this functor and on its right adjoint are given in Section 2.

In Section 3 we recall the definition of HOM_{G/K,R}(M, N) and we investigate its relation to HOM_{G/H,R}(M, N) in connection with the finite topology on Hom_{R}(M, N). Our situation gives rise to two composite functor:

HOM_{G/H,R}(M, −) ∘ U_{G/H}^{G/K}: (G/K, R)-Gr → (G/H, E)-Gr,

and

U_{G/H}^{G/K} ∘ HOM_{G/K,R}(M, −): (G/K, R)-Gr → (G/H, E)-Gr.

Loosely speaking we ask when these functors are equal. Our main result in Section 4 generalize [GMN, Th. 3.4] and states that if H/K is infinite, then the functors are equal if and only if M is a small R-module.

In this paper rings are always associative with unit element, and modules are unitary and left. Besides the above mentioned papers, we refer to [NRV] for general facts on graded rings and modules. Some related topics are also discussed in [I].

2. The grade forgetting functor and its adjoint

In this section we provide the details on the properties of the grade forgetting functor. Let R = ⨁_{g∈G} R_{g} be a G-graded ring as in the introduction, and fix two subgroups

K ≤ H of G.

2.1. The grade forgetting functor

U = U_{G/H}^{G/K}: (G/K, R)-Gr → (G/H, R)-Gr

is defined as follows: for M = ⨁_{x∈G/K} M_{x} ∈ (G/K, R)-Gr we have U(M) = M̄ = ⨁_{y∈G/H} M̄_{y}, where M̄ = M (as R-module), and M̄_{y} = ⨁_{x⊆y} M_{x} for all y ∈ G/H, and obviously, U(f) = f for every morphism f: M → M′ in (G/K, R)-Gr.

2.2. There is functor in the opposite direction

F = F_{G/H}^{G/K}: (G/H, R)-Gr → (G/K, R)-Gr

defined as follows: for N = ⨁_{y∈G/H} N_{y} ∈ (G/H, R)-Gr we have

F(N) = Ñ = ⨁_{x∈G/K} Ñ_{x}, Ñ_{x} = N_{xH},

with multiplication by scalars given by r_{g}ñ_{x} = r_{g}n_{y} ∈ N_{gx}, where y = xH, ñ_{x} = n_{y} ∈ N_{y}, r_{g} ∈ R_{g}, g ∈ G.

If f: N → N′ is morphism in (G/H, R)-Gr, then f̃ = F(f): Ñ → Ñ′ is given by f̃(ñ_{x}) = f(n_{y}) ∈ Ñ_{x} = N_{y}, with y = xH and ñ_{x} = n_{y} as above.

2.3. Lemma. With the above notation we have:

a) F = F_{G/H}^{G/K} is a right adjoint of U = U_{G/H}^{G/K}

b) If H/K is finite then F is also a left adjoint of U.

Proof. Let M ∈ (G/K, R)-Gr and N ∈ (G/H, R)-Gr. We define the functorial isomorphism

Φ_{M,N}: Hom_{(G/H,R)-Gr}(U(M), N) → Hom_{(G/K,R)}(M, F(N)),

by letting, for f: U(M) → N and m_{x} ∈ M_{x},

Φ_{M,N}(f)(m_{x}) = f(m_{x}) ∈ F(N)_{x} = N_{xH}.

Clearly, Φ_{M,N} is a well defined group homomorphism.

Define further

Ψ_{M,N}: Hom_{(G/K,R)-Gr}(M, F(N)) → Hom_{(G/H,R)}(U(M), N),

by letting, for g: M → F(N) and m_{x} ∈ M_{x} = U(M)_{xH},

Ψ_{M,N}(g)(m_{x}) = g(m_{x}) ∈ F(N)_{x} = N_{xH}.

Again Ψ_{M,N} is well defined and it is obvious that Ψ_{M,N} = Φ_{M,N}^{−1} .

b) Assume that H/K is finite, and for M and N as above define

Θ_{N,M}: Hom_{(G/K,R)-Gr}(F(N), M) → Hom_{(G/H,R)}(N, U(M)),

and for f: F(M) → N and n_{y} ∈ N_{y},

Θ_{N,M}(f)(n_{y}) = ∑_{\begin{matrix}x∈G/K \\ x⊆y \end{matrix}} f(ñ_{x}),

with ñ_{x} = n_{y} ∈ F(N)_{x}.

Conversely, define

Γ_{N,M}: Hom_{(G/H,R)-Gr}(N, U(M)) → Hom_{(G/K,R)}(F(N), M),

by letting, for g: N → U(M) and ñ_{x} = n_{y} ∈ F(N)_{x} = N_{xH}, y = xH

Γ_{N,M}(g)(ñ_{x}) = g(n_{x})_{x},

where g(ñ_{x}) is the x-th component of g(ñ_{x}) ∈ ⨁_{x∈G/K} M_{x}. Again it is easy to check that Γ_{N,M} = Θ_{N,M}^{−1} .

2.4. Remark. a) The unit ζ of the adjoint pair (U, F) is defined by

ζ_{M}: M → F(U(M)), ζ_{M}(m_{x}) = m_{x} ∈ F(U(M))_{x},

for all x ∈ G/K, m_{x} ∈ M_{x}. The counit ξ is given by

ξ_{N}: U(F(N)) → N, ξ_{N}(ñ_{x}) = ñ_{x} ∈ N_{xH},

for all ñ_{x} = n_{y} ∈ N_{y}, x ∈ G/K, y = xH.

b) If H/K is finite then the unit η_{N} of the adjoint pair (F, U) is defined by

η_{N}: N → U(F(N)), η_{N}(n_{y}) = ∑_{\begin{matrix}x∈G/K \\ x⊆y \end{matrix}} ñ_{x},

where ñ_{x} = n_{y} ∈ N_{y}. The counit μ_{M} is defined by

μ_{M}: F(U(M)) → M, μ_{M}(^{x′}m_{x}) = \{ \begin{matrix}m_{x}, & if x = x′ \\ 0, & if x ≠ x′, \end{matrix}

where x, x′ ∈ G/K, xH = x′H and ^{x′}m_{x} ∈ F(U(M))_{x′}.

Observe that μ_{M} ∘ ζ_{M} = id_{M} and ξ_{N} ∘ η_{N} =| H/K | id_{N}.

c) Recall that if (N_{i})_{i∈I} is a family of objects of (G/H, R)-Gr then the direct sum of this family is ⨁_{i∈I} N_{i} with G/H -grading given by (⨁_{i∈I} N_{i})_{y} = ⨁_{i∈I}(N_{i})_{y}. It follows that (G/K, R)-Gr is an AB3 category, and by its construction, it easily follows that F_{G/H}^{G/K} preserves direct sum.

3. The finite topology on HOM_{G/K,R}(M, N)

3.1. Let R be a G-graded ring, K a subgroup of G, M = ⨁_{g∈G} M_{g} ∈ R-Gr, and N = ⨁_{x∈G/K} N_{x} ∈ (G/K, R)-Gr. By [M, 2.9], for each x ∈ G/K, the set

HOM_{G/K,R}(M, N)_{x} = {f ∈ Hom_{R}(M, N) | f(M_{g}) ⊆ N_{gx} for all g ∈ G}

is an additive subgroup of Hom_{R}(M, N). Moreover the sum

HOM_{G/K,R}(M, N) = ∑_{x∈G/K} HOM_{G/K,R}(M, N)_{x}

is direct. Remark that we also have the equality

HOM_{G/K,R}(M, N)_{x} = Hom_{(G/^{x}K,R)-Gr}(U_{G/^{x}K}^{G/1}(M), N(x)),

where ^{x}K = xKx^{−1} , U_{G/^{x}K}^{G/1} : R-Gr → (G/^{x}K, R)-Gr is the grade-forgetting functor and N(x) is an object of (G/^{x}K, R)-Gr defined by

N(x) = N, N(x)_{y} = N_{yx} for every y ∈ G/^{x}K.

Observe (see also [M, 2.7.]) that the construction of N(x) actually gives an equivalence of categories

S^{x}: (G/K, R)-Gr → (G/^{x}K, R)-Gr, S^{x}(N) = N(x),

called the x-th suspension functor.

In particular, we have that End_{R}(M) contains a G-graded subring

END_{R}(M) = ⨁_{g∈G} END_{R}(M)_{g},

with

END_{R}(M)_{g} = {f ∈ End_{R}(M) | f(M_{h}) ⊆ M_{hg} for all h ∈ G},

and HOM_{G/K,R}(M, N) becomes a G/K -graded E -module where E = END_{R}(M)^{opp}.

The aim of this section is to establish the relationship between HOM_{G/K,R}(M, N) and Hom_{R}(M, N) in topological terms.

3.2. Let A be a Grothendieck category and let M, N be two objects of A. Recall that the finite topology on Hom_{R}(M, N) is defined by giving a basis for the filter of the neighborhoods of 0 as follows:

F(0) = {V (0, X) | X ranges over the finitely generated subobjects of M},

where

V (0, X) = V_{M,N}(0, X) = {f ∈ Hom_{A}(M, N)|X ⊆ Ker f}.

In this way, Hom_{A}(M, N) becomes a topological abelian group, which is actually a subspace of N^{M} endowed with the product topology, each of the components having the discrete topology.

If A is locally finitely generated (that is, it has a set of finitely generated generators or, equivalently, every object is the sum of its finitely generated subobject), then Hom_{A}(M, N) is a complete Hausdorff topological space in the finite topology (see [GMN, Section 2]). In particular this hold if A = R-mod. In this case, a filter of neighborhoods of 0 is

{V (0, X) | X is a finite subset of M}.

Indeed, if V (0, {x_{1}, …, x_{n}}) is a neighborhood of 0 in the finite topology, defined in this way, then V (0, {x_{1}, …, x_{n}}) = V (0, ⟨x_{1}, …, x_{n}⟩) is a neighborhood of 0 in the finite topology; conversely, V (0, X) = V (0, {x_{1}, …, x_{n}}) where {x_{1}, …, x_{n}} generates the submodule X of M.

3.3. Returning to the case when X ∈ R-Gr and N ∈ (G/K, R)-Gr we have that HOM_{G/K,R}(M, N) ⊆ Hom_{R}(M, N) is a topological space with the induced topology and this is precisely the finite topology. Indeed, if

V′(0, {x_{1}, …, x_{n}}) = {f ∈ HOM_{G/K,R}(M, N) | f(x_{1}) = … = f(x_{n})}

is a neighborhood of 0 in the finite topology on HOM_{G/K,R}(M, N), then

V′(0, {x_{1}, …, x_{n}}) = V (0, {x_{1}, …, x_{n}}) ∩ HOM_{G/K,R}(M, N),

where

V (0, {x_{1}, …, x_{n}}) = {f ∈ Hom_{R}(M, N) | f(x_{1}) = … = f(x_{n})}.

Moreover, for every f ∈ HOM_{G/K,R}(M, N) we have that

V′(f, {x_{1}, …, x_{n}}) = f + V′(0, {x_{1}, …, x_{n}}).

Thus HOM_{G/K,R}(M, N) is a Hausdorff topological abelian group in the finite topology.

3.4. Proposition. For any x ∈ G/K, Hom_{G/K,R}(M, N)_{x} is a closed subset of Hom_{R}(M, N) in the finite topology.

Proof. If f ∈ Hom_{R}(M, N) belongs to the closure of HOM_{G/K,R}(M, N)_{x}, and m_{g} ∈ M_{g}, then there is ϕ ∈ HOM_{G/K,R}(M, N) such that ϕ ∈ V (f, m_{g}), hence f(m_{g}) = ϕ(m_{g}). But ϕ(M_{g}) ⊆ N_{gx}, so we have that f(m_{g}) = ϕ(m_{g}) ∈ N_{gx}. Consequently, f(M_{g}) ⊆ N_{gx} for any g ∈ G, hence f ∈ HOM_{G/K,R}(M, N)_{x}.

3.5. Recall that if X is a topological space, then the family {x_{i} | i ∈ I} is called summable to x ∈ X if for every neighborhood V of x there is a finite subset J_{V} of I such that for each finite subset J of I containing J_{V} we have ∑_{i∈J} x_{i} ∈ V . In this case we write ∑_{I∈I} x_{i} = x.

We claim that if X = Hom_{R}(M, N), then it is enough to consider only neighborhoods of the type V (f, {m}) where f ∈ Hom_{R}(M, N) and m is a homogeneous element of M. Indeed, let {f_{i} | i ∈ I} be a family in Hom_{R}(M, N) with the property that for every neighborhood V (f, m_{g}) with m_{g} ∈ M_{g}, there is a subset J_{0} of I such that for every finite subset J of I containing J_{0} we have ∑_{i∈J} f_{i} ∈ V (f, m_{g}). Now let m = m_{1}+…+m_{n} ∈ M where m_{1}, …, m_{n} are homogeneous and let J_{1}, …, J_{n} be finite subsets such that for every j ∈ {1, …, } and for every finite subset J of I with J_{j} ∈ J we have ∑_{i∈J} f_{i} ∈ V (f, m_{j}), that is ∑_{i∈J} f_{I}(m_{j}) = f(m_{j}). If we let J_{0} = ⋃_{j=1}^{n} J_{j}, then (∑_{i∈J} f_{i})(m_{j}) = f(m_{j}), hence ∑_{i∈J} f_{i} ∈ V (f, m) for every finite subset J of I containing J_{0} . Finally let V = V (f, {m_{1}, …, m_{k}}) be a neighborhood of f. Then there are the finite subsets J′_{1}, …, J′_{k}, of I such that for every l ∈ {1, …, k} and for every finite subset J of I containing J′_{l} we have (∑_{i∈J} f_{i})(m_{l}) = f(m_{l}). If we set J_{V} = ⋃_{l=1}^{k} J′_{l}, then for every finite subset J of I containing J_{V} we have ∑_{i∈J} f_{i} ∈ V , hence ∑_{i∈I} f_{i} = f, and the claim is proved.

The goal of this section is to show that Hom_{R}(M, N) is the completion of its subgroup HOM_{G/H,R}(M, N), in the finite topology and to and to examine the relationship between HOM_{G/H,R}(M, N) and HOM_{G/K,R}(M, N), where K ≤ H are subgroups of G.

3.6. Let f ∈ Hom_{R}(M, N), g ∈ G and x ∈ G/H. We define the abelian groups homomorphism

f_{x}: M = ⨁_{g∈G} M_{g} → N

with components

f_{x}^{g}: M_{g} → N, f_{x}^{g}(m_{g}) = f(m_{g})_{gx} for all m_{g} ∈ M_{g},

where f(m_{g})_{gx} is the homogeneous component of f(m_{g}) ∈ N = ⨁_{y∈G/K} N_{y} belonging to N_{gx}.

If r_{h} ∈ R_{h}, h ∈ G, then r_{h}m_{g} ∈ M_{hg}, so

f_{x}(r_{h}m_{g}) = f_{x}^{hg}(r_{h}m_{g}) = f(r_{h}m_{g})_{hgx} = (r_{h}f(m_{g}))_{hgx}.

Since every element of N has a unique decomposition into homogeneous components, it follows that f_{x}(r_{h}m_{g}) = r_{h}f_{x}(m_{g}). Consequently, f_{x} is R-linear. Further, by definition we have that f_{x} ∈ HOM_{G/K,R}(M, N)_{x}.

Now we may state the main result of this section.

3.7. Theorem. If M ∈ R-Gr and N ∈ (G/K, R)-Gr, then with the above notations the following statements hold:

a) The family {f_{x} | x ∈ G/K} is summable to f in the finite topology, and the components f_{x} are uniquely determined by the properties f_{x} ∈ HOM_{G/K,R}(M, N)_{x} and ∑_{x∈G/K} f_{x} = f.

b) Hom_{R}(M, N) is the completion of HOM_{G/K,R}(M, N) in the finite topology.

Proof. a) By (3.4) we may restrict to the case of neighborhoods of the form V (f, m_{g}) where m_{g} ∈ M_{g} for some g ∈ G. Let f(m_{g}) = n_{x_{1}} + … + n_{x_{k}} with n_{x_{j}} ∈ N_{x_{j}} be the decomposition of f(m_{g}) into homogeneous components. We set J_{0} = {x_{1}, … x_{k}}. Then J_{0} is a finite subset of G/H and f(m_{g}) = ∑_{x∈J_{0}} f_{x}(m_{g}), hence ∑_{x∈J_{0}} f_{x} ∈ V (f, m_{g}). Since for each x ∈ G/K \ J_{0} we have f(m_{g}) = 0, it follows that for every finite subset J of I containing J_{0} , f(m_{g}) = (∑_{x∈J} f_{x}(m_{g}). Therefore ∑_{x∈J} f_{x} ∈ V (f, m_{g}), and consequently, ∑_{x∈G/K} f_{x} = f in the finite topology.

For the uniqueness, we assume that {g_{x} | x ∈ G/K} is another family of morphisms such that g_{x} ∈ HOM_{G/K,R}(M, N)_{x} and ∑_{x∈G/K} g_{x} = f. If f_{x_{0}} ≠ g_{x_{0}} for some x_{0} ∈ G/K, then there exits m_{g} ∈ M_{g} such that f_{x_{0}} ≠ g_{x_{0}} . We consider the neighborhood V (f, m_{g}) of f in the finite topology. Then we may find the finite subset J_{0} ∈ P_{f}(G/K) (where P_{f}(G/K) denotes the set of all finite subsets of G/K) with the property that for every J ∈ P_{f}(G/K) for which J_{0} ⊆ J we have ∑_{x∈G/K} f_{x} ∈ V (f, m_{g}) and ∑_{x∈G/K} g_{x} ∈ V (f, m_{g}). If we set J = J_{0} ∪ {x_{0}}, then J ∈ P_{f}(G/K) and ∑_{x∈J} f_{x}(m_{g}) = f(m_{g}) = ∑_{x∈J} g_{x}(m_{g}). By the uniqueness of the decomposition into homogeneous elements it follows that f_{x_{0}}(m_{g}) = g_{x_{0}}(m_{g}).

b) Let f be an element of Hom_{R}(M, N). With the above notations, for some J ∈ P_{f}(G/K) we have

∑_{x∈J} f_{x} ∈ ⨁_{x∈G/K} HOM_{G/K,R}(M, N)_{x} = HOM_{G/K,R}(M, N),

and a) implies that HOM_{G/K,R}(M, N) ∩ V (f, m) ≠ 0. Hence HOM_{G/K,R}(M, N) is dense in Hom_{R}(M, N) in the finite topology. But Hom_{R}(M, N) is a complete Hausdorff topological space in the finite topology, hence it is the completion of HOM_{G/K,R}(M, N).

3.8. Corollary. a) HOM_{G/K,R}(M, N) is dense in HOM_{G/H,R}(M, N) in the finite topology.

b) The subgroup HOM_{G/K,R}(M, N) consists of all R-homomorphisms f: M → N belonging to HOM_{G/H,R}(M, N) and satisfying the following property: for every g ∈ G there is a finite set F_{gH} ⊆ H/K such that for every σ ∈ G,

f_{gH}(M_{σ}) ⊆ ∑_{x∈F_{gH}} N_{σgx}.

c) In particular, HOM_{G/H,R}(M, N) = HOM_{G/K,R}(M, N) if either of the following conditions is satisfied:

(1) H/K is a finite set.

(2) M is a finitely generated R-module.

(3) supp(M) = {g ∈ G | M_{g} ≠ 0} and supp(N_{y}) = {x ∈ G/K | x ⊆ y, N_{x} ≠ 0} are finite sets.

Proof. a) is obvious by the theorem, since HOM_{G/K,R} ⊆ Hom_{G/H,R} ⊆ Hom_{R}(M, N).

b) Let [G/H] be a system of representatives for the left cosets of H in G, and [H/K] a system of representatives for the left cosets of K in H. If f ∈ Hom_{R}(M, N), then

f = ∑_{g∈[G/H]} f_{gH},

where f_{gH} ∈ HOM_{G/H,R}(M, N)_{gH}, and f belongs to HOM_{G/H,R}(M, N) if and only if the sum is finite. In this case we have that

f = ∑_{g∈[G/H]} ∑_{h∈[H/K]} f_{ghK} = ∑_{h∈[H/K]} f̃_{hK},

where f_{ghK} ∈ HOM_{G/K,R}(M, N)_{ghK}, f̃_{hK} = ∑_{g∈[G/H]} f_{ghK}, and f belongs to HOM_{G/K,R}(M, N) if and only if for any g ∈ [G/H], ∑_{h∈[H/K]} f_{ghK} is a finite sum. Finally, use that f_{ghK}(M_{σ}) ⊆ N_{σghK} for any σ ∈ G.

c) Case (1) is obvious by b). If M is finitely generated, then by [M, Corollary 2.11.b)] we even have that HOM_{H/K,R}(M, N) = Hom_{R}(M, N). For (3), let supp(M) = {σ_{1}, …, σ_{s}}, supp(N_{gH}) = {gh_{1}K, …, gh_{t}K}, and let F = F_{gH} = {σ_{i}^{−1}gh_{j}K | i = 1, …, s, j = 1, …, t}. If f ∈ HOM_{G/H,R}(M, N)_{gH} and σ ∈ G, then for σ ∉ supp(M), f(M_{σ}) = 0. If σ ∈ supp(M), then f(M_{σ}) ⊆ N_{gH} = ∑_{x∈F} N_{σx}.

4. Small graded modules

4.1. Let A be an abelian category satisfying AB3, and let M be an object of A. M is called small if the functor Hom_{A}(M, −): A → Ab preserves direct sum or, equivalently, preserves denumerable direct sum [S, p. 134]. This is equivalent to the fact that any morphism f: M → ⨁_{i∈ℕ} X_{i} factors trough a finite coproduct ⨁_{i∈F} X_{i}, where F a finite subset of ℕ. More generally we say that M is N -small if Hom_{A}(M, −): A → Ab preserves (denumerable) direct sum of copies of N (for N = M one gets the concept of a self-small object). We will use the following results.

4.2. Proposition. [GMN, Proposition 1.1] Let A be an AB3 category and M an object of A. Then M is small if and only if M is N -small for every object N of A.

4.4. Theorem. [GMN, Theorem 1.3.] Let A and B be AB3 categories, M ∈ A and N ∈ B. Let U: A → B be a left adjoint of the functor F: B → A and assume that F preserves direct sum. Then the following assertion hold:

a) M is F(N)-small in A if and only if U(M) is N -small in B.

b) If M is small in A then U(M) is small in B.

4.5. The connection between small objects and finite topology is as follows (cf. [GMN, pp. 3176-3177]). If the finite topology of Hom_{A}(M, N) is discrete, where A is a Grothendieck category, then M is N -small. The converse is true if we assume in addition that A being locally finitely generated and the finite topology of Hom_{A}(M, N) being first countable. If A is a locally finitely generated Grothendieck category and M ∈ A is countably generated, then M is N -small if and only if the finite topology of Hom_{A}(M, N) is discrete.

Now we return to our context, and let

U = U_{G/H}^{G/K}: (G/K, R)-Gr → (G/H, R)-Gr

be the grade forgetting functor introduced in Section 2 and

F = F_{G/H}^{G/K}: (G/H, R) → (G/K, R)-Gr

its right adjoint.

4.6. Proposition. Let R be a G-graded ring and K ≤ H subgroups of G. If M and N are two objects of (G/K, R)-Gr, then the following statements hold:

a) M is small in (G/H, R)-Gr if and only if M is small in (G/K, R)-Gr.

b) M is U(N)-small in (G/H, R)-Gr if and only if M is F(U(N))-small in (G/K, R)-Gr.

Proof. a) The necessity is obvious since we have that Hom_{(G/K,R)-Gr}(M, N) is a subset of Hom_{(G/H,R)-Gr}(M, N) for every N ∈ (G/K, R)-Gr. The sufficiency follows by Theorem 4.4.b), applied to the functor U whose right adjoint F preserves direct sum.

b) By Theorem 4.2.a), we have that M = U(M) is U(N)-small in (G/H, R)-Gr if and only if M is F(U(N))-small in (G/K, R)-Gr.

4.7. Remark. If M is a G-graded R-module, then M may be regarded as an object of (G/K, R)-Gr via the functor

U_{G/K}^{G/1} : R-Gr → (G/K, R)-Gr.

If N is a G/K -graded R-module, then statement b) of the above proposition becomes: The object M of R-Gr is U_{G/H}^{G/K}(N)-small in (G/H, R)-Gr if and only if M is F(U(N))-small in (G/K, R)-Gr.

4.8. Lemma. Let M ∈ R-Gr and N ∈ (G/K, R)-Gr. If M is U(N)-small in (G/H, R)-Gr, then

HOM_{G/H,R}(M, N) = HOM_{G/H,R}(M, N).

Proof. Let y = G/H and f ∈ Hom_{G/H,R}(M, N)_{y}. As in (3.6), for each x ∈ H/K such that x ⊆ y, we associate to f a morphism f_{x} ∈ HOM_{G/K,R}(M, N)_{x} defined by f_{x}(m_{g}) = f(m_{g})_{gx}, for any g ∈ G and m_{g} ∈ M_{g}.

For each x ∈ H/K, x ⊆ y denote by N^{x} a copy of N, and define

ϕ: M → ⨁_{\begin{matrix}x∈G/K \\ x⊆y \end{matrix}} N^{x}, ϕ(m) = (f_{x}(m))\begin{matrix}x∈G/K \\ x⊆y \end{matrix}, ∀ m ∈ M.

We clearly have that ϕ ∈ HOM_{G/H,R}(M, N)_{y} = Hom_{(G/^{y}H,R)-Gr}(M, ⨁_{x} N^{x}(y)), where recall that N(y) is the y -suspension of N introduced in (3.1). Since the suspension functor S^{x}: (G/K, R)-Gr → (G/^{x}K, R)-Gr of (3.1) is an equivalence, we have that M is N(y) small in (G/^{y}H, R)-Gr, hence there are x_{1}, …, x_{n} ∈ H/K, x_{i} ⊆ y, i = 1, …, n, such that ϕ(M) ⊆ ⨁_{i=1}^{n} N^{x_{i}}(y). It follows that f_{x} = 0 for x ∉ {x_{1}, …, x_{n}}, and therefore f = ∑_{i=1}^{n} f_{x_{i}}. Consequently, f ∈ HOM_{G/H,R}(M, N).

We finally arrived to the main result of this paper.

4.9. Theorem. Let R be a G-graded ring, M a G-graded R-module, and let K ≤ H be two subgroups of G such that H/K is infinite. Then the following statements are equivalent:

(i) M is small in (G/K, R)-Gr (or equivalently, in R-mod, or in R-Gr, or in (G/H, R)-Gr).

(ii) HOM_{G/H,R}(M, N) = HOM_{G/K,R}(M, N) for every G/K -graded R-module N. Proof. The fact that (i) implies (ii) follows immediately by Proposition 4.2 and Lemma 4.8.

Conversely, let M = ⨁_{g∈G} M_{g} ∈ R-Gr such that HOM_{G/H,R}(M, N) = Hom_{R}(M, N) for every N ∈ (G/H, R)-Gr. Let

f: M → ⨁_{i∈ℕ} X_{i}

be a morphism in R-Gr. We are going to show that f factorizes through a finite subcoproduct of ⨁_{i∈ℕ} X_{i}.

Let X = ⨁_{i∈ℕ} X_{i}, and let N = ⨁_{g∈G} X(g), where X(g) ∈ R-Gr is the g -suspension of N. It follows that N ≃ N(g) in R-Gr, for every g ∈ G. Let [H/K] = {h_{i} | i ∈ I} be a system of representatives for the left cosets of K in H. Since H/K is infinite, we may assume that ℕ is a subset of I. It follows that there is a monomorphism

v: N^{(ℕ)} → ⨁_{i∈I} N(h_{i}).

Let u_{1} : X(1) = X → N = ⨁_{g∈G} X(g) and q_{i}: X_{i} → X = ⨁_{i∈ℕ} X_{i} be the canonical injections, where i ∈ ℕ. Consider the composition u_{1} ∘ q_{i}: X_{i} → N, and let u be the monomorphism

u := ⨁_{i∈ℕ}(u_{1} ∘ q_{i}): ⨁_{i∈ℕ} X_{i} → ⨁_{i∈ℕ} N

of G-graded R-modules. Let further

ϕ := v ∘ u ∘ f: M → ⨁_{i∈I} N(h_{i}),

so ϕ is also a monomorphism in R-Gr. Let

t: ⨁_{i∈I} N(h_{i}) → N, t((n(h_{i})_{i∈I}) = ∑_{i∈I} n(h_{i}),

and let ψ = t ∘ ϕ: M → N. We clearly have that ψ belongs to Hom_{(G/H,R)-Gr}(M, N) = HOM_{G/H,R}(M, N)_{H}. By assumption, we have that

HOM_{G/H,R}(M, N)_{H} = ⨁_{i∈I} HOM_{G/H,R}(M, N)_{h_{i}K},

so there are elements h_{1}, …, h_{n} ∈ [H/K] and ψ_{i} ∈ HOM_{G/H,R}(M, N)_{h_{i}K}, i = 1, …, n, such that ψ = ∑_{i=1}^{n} ψ_{i} (we may change the indices of elements in [H/K], if needed). It follows that for every g ∈ G, ψ(M_{g}) ⊆ N_{gh_{1}}+⋯+N_{gh_{n}}. Further, if σ ∈ G and m_{σ} ∈ M_{σ} such that ϕ(m_{σ}) = (n(g))_{g∈G} with n(g) ∈ N_{σg}, then ψ(m_{g}) = ∑_{g∈G} n(g) ∈ N_{σh_{1}} + ⋯ + N_{σh_{n}}; therefore, n(g) = 0 for g ∉ {h_{1}, …, h_{n}}. We deduce that Im f ⊆ ⨁_{i=1}^{n} X_{i}, hence M is small.

4.10. Example. Let K ≤ H be two subgroups of a group G such that H/K is infinite, and let R be the group ring ℤG regarded as a G-graded ring in a natural way. If M = ⨁_{g∈G} R(g), then it is not difficult to see that HOM_{G/K,R}(M, M) is strictly included in HOM_{G/H,R}(M, M).

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